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Math Club

Thursday, March 14th, 2019

Time: 5:30 - 6:30 p.m Place: Jeffery Hall 118

Speaker: Richard Gottesman (Queen's University)

Title: Gaussian Integers and Sums of Two Squares.

Abstract: The Gaussian integers consist of the set of all complex numbers whose real and imaginary parts are both integers. We begin by exploring the number theory of the Gaussian integers. For example, we shall see why $3$ is a Gaussian prime but $5 = (2 +i)(2 - i)$ is not.

We will then show how to use the Gaussian integers to prove that if $p$ is a prime number which is one more than a multiple of $4$ then $p$ is a sum of two perfect squares. This proof is very striking and it generalizes to other number systems, such as the Hurwitz quaternions.

In honor of $\pi$ day, we must also mention that $\pi$ is equal to the average number of ways to write an integer as a sum of two squares.